The Kim-Pillay theorem for Abstract Elementary Categories

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Abstract

We introduce the framework of AECats (abstract elementary categories), generalizing both the category of models of some first-order theory and the category of subsets of models. Any AEC and any compact abstract theory (cat, as introduced by Ben-Yaacov) forms an AECat. In particular, we find applications in positive logic and continuous logic: the category of (subsets of) models of a positive or continuous theory is an AECat. The Kim-Pillay theorem for first-order logic characterizes simple theories by the properties dividing independence has. We prove a version of the Kim-Pillay theorem for AECats with the amalgamation property, generalizing the first-order version and existing versions for positive logic.

Original languageEnglish
Pages (from-to)1717-1741
Number of pages25
JournalJournal of Symbolic Logic
Volume85
Issue number4
Early online date30 Oct 2020
DOIs
Publication statusPublished - Dec 2020

Keywords

  • dividing
  • accessible category
  • simple theory
  • abstract elementary class
  • independence relation
  • abstract elementary category

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